Question

Let \( X_1, X_2, \dots, X_n \) be a random sample from a continuous distribution with the probability density function \( f(x) \). To test the hypothesis \( H_0: f(x) = e^{-x^2} \) against \( H_1: f(x) = e^{-2|x|} \), the rejection region of the most powerful size \( \alpha \) test is of the form, for some \( c>0 \),

• A. \( \sum_{i=1}^{n} (X_i - 1)^2 \geq c \)
• B. \( \sum_{i=1}^{n} (X_i - 1)^2 \leq c \)
• C. \( \sum_{i=1}^{n} |X_i| \geq c \)
• D. \( \sum_{i=1}^{n} |X_i - 1|^2 \leq c \)

Hint:

For hypothesis testing, the most powerful test is often derived using the likelihood ratio test. The rejection region is defined by comparing the test statistic to a critical value.

Answer 1

The Correct Option is C

Step 1: Likelihood Ratio Test. 
The rejection region for the most powerful test is determined by comparing the likelihood ratio for the two hypotheses. The likelihood ratio test is used to identify the test statistic. 
Step 2: Determine the form of the test statistic. 
For this problem, we can derive the test statistic by calculating the likelihood ratio and finding the critical region based on the distribution of the test statistic under the null hypothesis. 
Step 3: Conclusion. 
The rejection region for the test is given by \( \sum_{i=1}^{n} (X_i - 1)^2 \geq c \), which corresponds to the most powerful test.